Andrew,
For starters, perhaps this illustration will help:
The situation is confused because we have not been precise and consistent about our terms.
In the
other thread, first post, you defined "extension" as being measured from the sensor to some lens plane, and you asked where that plane was.
Measuring from the sensor, the reference plane in reversed orientation is indeed the one labeled "H", which is close to the mounting flange. In this case Magnification = (E/f)-1.
But in your
web page calculator, you are now using "extension" in a different way: the amount
added,
beyond infinity focus. That's the only way that "Magnification = E/f".
The user must have some way to measure this extension.
I casually and perhaps incorrectly assumed that you intended it to be measured from the camera lens flange to some point on the lens.
It happens, with a Canon EOS, that both styles of EL Nikkor 50 mm f/2.8 reach infinity focus in normal orientation when the shoulder of the mounting threads is roughly lined up with the front of the camera's lens mount.
So to a user measuring extensions from the camera's lens mount, the proper reference point on the lens in normal orientation is roughly the shoulder of the mounting threads, as I have it diagrammed. Around 0 mm of "extension" will give infinity focus.
When the lens is reversed, it cannot actually reach infinity focus because the lens would have to push into the camera body.
If one could push the lens into the camera body, then infinity focus would occur when the plane that I marked "Reversed" is lined up with the front of the camera lens mount. If you want 2X magnification, then you need that point to be 2*50 = 100 mm away from the front of the camera lens mount. Measuring 100 mm from the lens mount to any other reference point on the lens will give some different magnification. For example, measuring 100 mm from the lens mount to the shoulder of the normal mounting threads will give a magnification closer to 1.72X, not the desired 2X. Measured from the sensor to the shoulder can give the right result, but in that case the extension would have to be 150 mm ((mag+1)*50), not 100 mm.
If you want to use the H and H' planes, that's fine, but then you'll have to define extension=(magnification+1)*focalLength, and measure from the focal plane.
An alternative approach is to just say
1/f = 1/o + 1/i will get me in the ballpark most of the time if I just use the sensor and physical center of the lens as reference points. If I need anything more accurate, then I'll measure the magnification and use the formulas based on magnification rather than focal lengths and extensions. If that's not accurate enough, well, then clearly things are getting complicated and it's safer and easier to just rely on experiment rather than theory.
That approach has served me very well for several decades. Despite that I now understand about principal planes and how they relate to reference points for measuring extensions, I generally do not think about all that when I'm going after a photo.
Wearing my scientist hat, I feel compelled to work theory until it matches observations. Wearing my photographer hat, I really don't care about lens theory beyond what it takes to get me in the right ballpark.
The beauty of analysis using principal planes H and H' is that it gives results that are accurate within measurement error. The ugliness is that there's a big learning curve to use them correctly, and if you use them not correctly you can easily get results that are worse than not using them at all.
I notice that
Lefkowitz doesn't even index the term "principal plane", and I can't recall seeing it mentioned in his text. Regarding reversal, what he says is this (page 68 ):
And if the lens has been reversed at high magnification to improve image resolution, the exact image size will be anywhere from 20-100 percent smaller or greater than calculated from the aforementioned formula. [E.D = m x F.L.] The exact amount depends on lens design, focal length, magnification, reversing-ring thickness, and even how deeply the lens is recessed in its mount. The best thing to do is to try it, and if an exact magnification figure is required, use the ruler technique...
For most beginners working with bellows, I'm inclined to recommend taking the approach outlined in the first quote above, "...use the sensor and physical center of the lens as reference points...", and just acknowledge that it's a ballpark calculation.
--Rik